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Differential Equations

Short Answer Type

71. Form the differential equation of the family of curves by eliminating arbitrary constants a and b.
y = a e^3x + be^-2x

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72. Form the differential equation of the family of curves by eliminating arbitrary constants a and b.
y = e^2x (a + b x)

111 Views

73. Form the differential equation of the family of curves by eliminating arbitrary constants a and b.
y = e^x (a cosx + b sinx)

94 Views

74. Form a differential equation from the equation y = ae ^2x + be^{– 3x} , a , b being constants.

The given equation is
  y = ae ^2x + be^{– 3x} ...(1)
Differentiating both sides w.r.t.x, we get
   $dy over dx space equals space 2 ae to the power of 2 straight x end exponent minus 3 be to the power of negative 3 straight x end exponent$ ...(2)
Again differentiation both sides w.r.t.x, we get.
   $fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space 4 ae to the power of 2 straight x end exponent plus 9 be to the power of negative 3 straight x end exponent$ ...(3)
Multiplying (2) by 2 and subtracting from (3), we get,
   $fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx space equals space 15 space be to the power of negative 3 straight x end exponent$
$therefore space space space space space be to the power of negative 3 straight x end exponent space equals 1 over 15 open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx close parentheses$ ...(4)
Multiplying (2) by 3 and adding it to (3), we get,
   $fraction numerator straight d squared straight y over denominator dx squared end fraction plus 3 dy over dx space equals space 10 space straight a space straight e to the power of 2 straight x end exponent$
$therefore space space space space space space ae to the power of 2 straight x end exponent space equals space 1 over 10 open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction plus 3 dy over dx close parentheses$ ...(5)
From (1), (4) and (5), we get,
$straight y equals space 1 over 10 open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction plus 3 dy over dx close parentheses plus 1 over 15 open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 dy over dx close parentheses$

or    $30 space straight y space equals space 3 space fraction numerator straight d squared straight y over denominator dx squared end fraction plus 9 dy over dx plus 2 fraction numerator straight d squared straight y over denominator dx squared end fraction minus 4 dy over dx$
or    $5 fraction numerator straight d squared straight y over denominator dx squared end fraction plus 5 dy over dx space equals space 30 space straight y space space space space space or space space space fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx minus 6 straight y space equals space 0$
which is required differential equation.

163 Views

75. Form the differential equation of the following family of curves:
x y = A e^x + B e^–x + x²

106 Views

76. Find the differential equation of the family of curves y = ae^x + be^2x + ce^3x, where a, b. c are arbitrary constants.

160 Views

77.

Obtain the differential equation from the equation y = e^x (a cos 2x + b sin 2x). where a and b are arbitrary constants.

93 Views

78. Find the differential equation of the family of curves y = a sin (bx + c). a, b. c being arbitrary constants.

89 Views

Long Answer Type

79. Form the differential equation of the family of circles having centre on x-axis and passing through the origin.

100 Views

Short Answer Type

80. Determine the differential equation that will represent the family of all circles having centres on the x-axis and radius unity.

101 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

74. Form a differential equation from the equation y = ae 2x + be– 3x , a , b being constants.

Long Answer Type

Short Answer Type

74. Form a differential equation from the equation y = ae ^2x + be^{– 3x} , a , b being constants.